I am trying to figure out how to prove that the following sequence converges:

with

Thoughts anyone?

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- Jun 3rd 2010, 07:18 AMmoemoeProof of Convergence for Recursive Sequence
I am trying to figure out how to prove that the following sequence converges:

with

Thoughts anyone? - Jun 3rd 2010, 07:53 AMmoemoe
- Jun 3rd 2010, 07:54 AMparkhid
I think we should say this :

and

it means :

and

also if we can unspot .

and

this is the proof. but to complete it

(Cool)

we should proof that . or

We Know L is 1 as here :

so the - Jun 3rd 2010, 07:57 AMPlato
- Jun 3rd 2010, 08:00 AMDefunkt
You can't assume convergence if you're trying to prove it...

@moemoe: The usual way to prove that this type of recursive sequences converge is to prove by induction that the sequence is monotone and bounded.

In your case, plug in the values for the first few terms to see whether the sequence is increasing or decreasing, and then prove it by induction. After you do that, prove that it is bounded (this can be done by induction as well).

EDIT: Plato ahead of me :< - Jun 3rd 2010, 03:53 PMmoemoe
So assuming

we have

Does anyone have a slick way of showing the sequence is bounded?

- Jun 3rd 2010, 03:59 PMmoemoe
Maybe define a new sequence

with

where for each and is decreasing by the same induction argument above?

Is there another simpler way? - Jun 3rd 2010, 04:08 PMDefunkt
There is a simpler way.

First, validate that .

Now, assume and use the induction hypothesis to prove that .

This will prove that the sequence is bounded by 2 (it doesn't mean that 2 is the limit, though..) - Jun 3rd 2010, 05:47 PMgalactus
Perhaps we can try this.

By the induction step it can be shown that

Since for all n,

and it is monotone increasing. - Jun 4th 2010, 01:59 AMchisigma
The sequence is defined recursively as...

(1)

The function is represented here...

http://digilander.libero.it/luposabatini/MHF63.bmp

It has a single zero at and because is less that the line crossing the x axis in with unity negative slope, any will produce a sequence converging at without oscillations...

Kind regards